Pythagoras' Theorem

What the theorem says

Take any right triangle — one with a 90° corner — and call the two shorter sides (the legs) a and b, and the longest side (opposite the right angle, the hypotenuse) c. The theorem states that the square of the hypotenuse always equals the sum of the squares of the other two sides.

It's not just an equation to memorize — it's a statement about area. If you build an actual square on each of the three sides of the triangle, the areas of the two smaller squares always add up exactly to the area of the square on the hypotenuse, no matter the triangle's proportions, as long as the angle is a right angle.

Use the tabs above to explore where this comes from: History traces it back nearly 4,000 years, Proof shows two different ways to see why it's true, Why It Matters covers where it shows up today, and Interactive Lab lets you drag a live triangle and watch the relationship hold.

A theorem older than its name

Despite carrying his name, Pythagoras almost certainly did not discover this relationship — and there is no surviving proof that he ever wrote one down himself. Historians note that the theorem was credited to him largely because his followers at Croton attributed all new results to their founder under strict vows of secrecy.

Why it still matters

Drag any two sides freely and the third is solved live using a² + b² = c². Or lock two sides and slide the third to watch the other two adjust automatically — the relationship holds in every direction, not just "find the hypotenuse."